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A127538 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k odd-length branches starting at the root (0<=k<=n).

%I A127538 #2 Mar 30 2012 17:36:12
%S A127538 1,0,1,1,0,1,0,4,0,1,3,3,7,0,1,3,22,6,10,0,1,16,43,50,9,13,0,1,37,175,
%T A127538 101,87,12,16,0,1,134,503,448,177,133,15,19,0,1,411,1784,1305,862,271,
%U A127538 188,18,22,0,1,1411,5887,4848,2524,1444,383,252,21,25,0,1,4747,20604
%N A127538 Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k odd-length branches starting at the root (0<=k<=n).
%C A127538 Row sums are the Catalan numbers (A000108). T(n,0)=A127539(n). Sum(k*T(n,k),k=0..n)=A127540(n).
%F A127538 G.f.=(1+z)/(1+z-z^2*C-tzC), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e A127538 T(2,2)=1 because we have the tree /\.
%e A127538 Triangle starts:
%e A127538 1;
%e A127538 0,1;
%e A127538 1,0,1;
%e A127538 0,4,0,1;
%e A127538 3,3,7,0,1;
%e A127538 3,22,6,10,0,1;
%p A127538 C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z^2*C-t*z*C): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A127538 Cf. A000108, A127539, A127540, A127541.
%K A127538 nonn,tabl
%O A127538 0,8
%A A127538 _Emeric Deutsch_, Mar 01 2007